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fourier

Fourier transform

Syntax

fourier(f)
fourier(f,transVar)
fourier(f,var,transVar)

Description

example

fourier(f) returns the Fourier Transform of f. By default, the function symvar determines the independent variable, and w is the transformation variable.

example

fourier(f,transVar) uses the transformation variable transVar instead of w.

example

fourier(f,var,transVar) uses the independent variable var and the transformation variable transVar instead of symvar and w, respectively.

Examples

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Compute the Fourier transform of exp(-t^2). By default, the transform is in terms of w.

syms t
f = exp(-t^2);
ft_f = fourier(f)
ft_f =
pi^(1/2)*exp(-w^2/4)

Compute the Fourier transform of exp(-t^2-x^2). By default, symvar determines the independent variable, and w is the transformation variable. Here, symvar chooses x.

syms t x
f = exp(-t^2-x^2);
fourier(f)
ans =
pi^(1/2)*exp(- t^2 - w^2/4)

Specify the transformation variable as y. If you specify only one variable, that variable is the transformation variable. symvar still determines the independent variable.

syms y
fourier(f,y)
ans =
pi^(1/2)*exp(- t^2 - y^2/4)

Specify both the independent and transformation variables as t and y in the second and third arguments, respectively.

fourier(f,t,y)
ans =
pi^(1/2)*exp(- x^2 - y^2/4)

Compute the following Fourier transforms. The results are in terms of the Dirac and Heaviside functions.

syms t w
fourier(t^3, t, w)
ans =
-pi*dirac(3, w)*2i
syms t0
fourier(heaviside(t - t0), t, w)
ans =
exp(-t0*w*1i)*(pi*dirac(w) - 1i/w)

Specify parameters of the Fourier transform.

Compute the Fourier transform of f using the default values of the Fourier parameters c = 1, s = -1. For details, see Fourier Transform.

syms t w
f = t*exp(-t^2);
fourier(f,t,w)
ans =
-(w*pi^(1/2)*exp(-w^2/4)*1i)/2

Change the Fourier parameters to c = 1, s = 1 by using sympref, and compute the transform again. The result changes.

sympref('FourierParameters',[1 1]);
fourier(f,t,w)
ans =
(w*pi^(1/2)*exp(-w^2/4)*1i)/2

Change the Fourier parameters to c = 1/(2*pi), s = 1. The result changes.

sympref('FourierParameters', [1/(2*sym(pi)), 1]);
fourier(f,t,w)
ans =
(w*exp(-w^2/4)*1i)/(4*pi^(1/2))

Preferences set by sympref persist through your current and future MATLAB® sessions. Restore the default values of c and s by setting FourierParameters to 'default'.

sympref('FourierParameters','default');

Find the Fourier transform of the matrix M. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars, fourier acts on them element-wise.

syms a b c d w x y z
M = [exp(x) 1; sin(y) i*z];
vars = [w x; y z];
transVars = [a b; c d];
fourier(M,vars,transVars)
ans =
[                 2*pi*exp(x)*dirac(a),     2*pi*dirac(b)]
[ -pi*(dirac(c - 1) - dirac(c + 1))*1i, -2*pi*dirac(1, d)]

If fourier is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size.

fourier(x,vars,transVars)
ans =
[ 2*pi*x*dirac(a), pi*dirac(1, b)*2i]
[ 2*pi*x*dirac(c),   2*pi*x*dirac(d)]

If fourier cannot transform the input then it returns an unevaluated call.

syms f(t) w
F = fourier(f,t,w)
F =
fourier(f(t), t, w)

Return the original expression by using ifourier.

ifourier(F,w,t)
ans =
f(t)

Input Arguments

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Input, specified as a symbolic expression, function, vector, or matrix.

Independent variable, specified as a symbolic variable. This variable is often called the "time variable" or the "space variable." If you do not specify the variable, then fourier uses the function symvar to determine the independent variable.

Transformation variable, specified as a symbolic variable, expression, vector, or matrix. This variable is often called the "frequency variable." By default, fourier uses w. If w is the independent variable of f, then fourier uses v.

More About

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Fourier Transform

The Fourier transform of the expression f = f(x) with respect to the variable x at the point w is

F(w)=cf(x)eiswxdx.

c and s are parameters of the Fourier transform. The fourier function uses c = 1, s = –1.

Tips

  • If any argument is an array, then fourier acts element-wise on all elements of the array.

  • If the first argument contains a symbolic function, then the second argument must be a scalar.

  • To compute the inverse Fourier transform, use ifourier.

References

[1] Oberhettinger F., "Tables of Fourier Transforms and Fourier Transforms of Distributions." Springer, 1990.

Introduced before R2006a

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